a level maths coursework

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Cambridge international as & a level mathematics (9709).

• Syllabus overview

Cambridge International A Level Mathematics develops a set of transferable skills. These include the skill of working with mathematical information, as well as the ability to think logically and independently, consider accuracy, model situations mathematically, analyse results and reflect on findings. 

Learners can apply these skills across a wide range of subjects and these skills equip them well for progression to higher education or directly into employment. 

At AS level, teachers can choose from three different routes to Cambridge International AS Level Mathematics: Pure Mathematics only, Pure Mathematics and Mechanics, or Pure Mathematics and Probability & Statistics. 

At A Level, teachers can also choose from different routes to Cambridge International A Level Mathematics depending on the choice of applied mathematics (Mechanics and/or Probability and Statistics).

Please note: From 2020 Cambridge International A Level Mathematics has two different routes: Pure Mathematics and Probability & Statistics only; or Pure Mathematics, Mechanics and Probability & Statistics.

The syllabus year refers to the year in which the examination will be taken.

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AQA A-Level Maths Revision Notes & Resources

Look at that, it’s almost time for you’re A-Level Maths exams! Before you know it, it’ll be exam season and you’ll be sitting one assessment after the other to complete your course and get your AQA A-Level Mathematics diploma. While the big bulk of revision ahead and the many exams that await may sound overwhelming, you shouldn’t panic. With the right revision plan and a structured approach before you know it, you’ll be ready for your exams. In order to help you through this busy period, here’s all you need to know about your upcoming exams and how we can help you succeed in them.

As you may have noticed A-Level Mathematics exams are fairly standardised. However, the many different examination boards do each take slightly different approaches to assessment and it’s good to keep this in mind. The AQA maths exams, for example, have a coursework component for one of their statistics modules, which is important to remember as you plan your revision.

About the Board

The Assessment and Qualification Alliance (AQA) exam board was born in 2000 after several exam boards merged in the previous decades. As one of the most prominent British examination boards, they offer a wide range of secondary school diplomas. Currently, the AQA A-Level Mathematics course only offers examinations in June each year. This means that the A-Level is graded in a linear, not a modular way i.e. that you take all of your exams not throughout the academic year but at the end of it.

How long will the course take to complete?

The AQA A-Level Mathematics course usually takes one or two academic years to complete. How long it will take exactly largely depends on whether you choose to take AS or A-Level Mathematics. If you take AS Level Mathematics, then the course is likely to take one year to complete. Meanwhile, if you opted for A-Level Mathematics then the full course will take you two years to finish. You are, of course, welcome to choose either AS or A-Level Maths. The choice is completely yours. However, you should bear in mind that many continuing education programs (be it apprenticeships, colleges or universities) require that you have a minimum of three A-Levels (and then as many AS Levels as you like).

Is any prior knowledge required?

The AQA board states that they have no specific requirements about prior courses and taking Mathematics exams at levels lower than this. However, as they specify, “[maths] is, inherently, a sequential subject,” they expect the A-Level Mathematics student to be up to date in terms of content in order to be able to begin the course. So you are generally recommended to have completed a prior course in maths, preferably GCSE Mathematics (or equivalent.

What will I study?

What you study will largely depend on whether you are taking AS or A-Level Mathematics. If you take AS Level Maths, then you must complete three modules to get your certificate. These must include the two core mathematics modules as well as one of the optional modules. The optional modules within the AQA AS Level Mathematics are Statistics 1, Statistics 2 (including coursework), Mechanics 1 and Decision-making 1.

Meanwhile, if you elect to take the full A-Level course then you will have to complete six modules to obtain your diploma. The obligatory modules will be the four core mathematics modules; on top of those, you will get to choose two optional modules. You can elect to take 2 modules in one of the 3 elective streams: mechanics, statistics and decision-making. Alternatively, you can take one of each – like Mechanics 1 and Statistics 1, or Mechanics 1 and Decision-Making 1, or Statistics 1 and Decision-Making 1. Be aware that your options may be limited by what is offered at your school.

What is the examination process like?

The AQA A-Level Mathematics exam process is fairly straight forward. If you’re taking AS Level maths, then you will take three exams (one for each of your modules). Each will last one hour and thirty minutes and count for a third of your final grade. The only exception is if you take the Statistics 2 module which also has a coursework component. In that case for this particular module, you will sit one exam for 1 hour and 15 minutes which will count for 25% of your final grade. In addition to that, you will complete coursework which will count for 8.3% of your final grade.

However, if you take the full A-Level Mathematics course then you will six exams, one for each of your six modules. Each of the exams will last 1 hour and 30 minutes and count for a sixth of your final grade. Again, if you take the Statistics 2 module then your assessment will change slightly. You will sit the 1 hour and 15 minutes long exam which will count for 12.5% of your final grade and you will complete the compulsory coursework which will count for just over 4% of your final grade.

Revision and practice are key to succeeding in your AQA A-Level Mathematics exams. Therefore, you are recommended to spend about three to four hours revising each day. While there are many important formulas, concepts, and rules to learn in maths, revision is most definitely not all about using just your textbook. In fact, research has shown that students with more diverse revision approaches tend to retain information more easily. Spice up your revision with tools like flashcards to remember your formulas and equations. And, most importantly in preparing for any maths exams practice really does make perfect. You should be making use of as many past exam papers as you can find. These will give you a great understanding of what types of questions are out there, and practicing using the rules and equations you’ve learned will help you remember. Of course, finding all those materials and structuring them isn’t easy. This is why we at A-Level Maths have compiled a whole range of quizzes, past papers, and flashcards for you to use in your revision. Take a look and get started today!

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C3 Coursework - different methods of solving equations.

C3 Coursework – Numerical Solutions

Decimal Search

There a numerous ways to solve a problem and in finding the unknown. Some methods give you the exact and precise answer but usually are harder and more complex. The Decimal search method enables you to get a very close approximate to the real solution but more easily.

The way this method works is by looking between two numerical values (for example 1 and 2) and then

As a demonstration in applying this method, I will be attempting to solve this equation using the Decimal Search method and going through the method step by step:

Below is what this function looks when plotted on a graph:

We know that the solution for F(x) = 0 is the point on the X axis where the sign changes from a positive to a negative.

So if we zoom in a little bit further, from this graph we can tell where the solution lies, somewhere between 0 and 10

Now that we know the solution is roughly between these two values, I will use excel to solve the problem with firstly taking increments in x, the size of 1. So when I substitute the incremented values of x between -10 and 0 into the equation, I get the following results:

You can tell that the sign changed between -3 and -4. So I set these as my initial values. The fact that the solution lies between -3 and -4 can also be seen in the graph:

So next, I check with increments of 0.1 in x. I again substitute the values in to the equation and tabulate the results and look for where the sign change occurs. I repeat this until I get down to increments the size of 0.00001

From the last table, you can see

that the solution lies between

-3.52719 and -3.52720

From the graph, you can see that the solution lies slightly to towards to the -3.52720 side.

The answer is between -3.52720 and -3.52719, which can be re-written as 3.527195 ± 0.00005

Failure of the Decimal Search Method

This method does not always work. In the case of F(x) = (x-4) 4  you cannot use this, method to find the roots at this does not cross the X axis. It merely touches it. So because this method relies on the line of the graph crossing the X +axis and the value changing signs, it is impossible to solve this with this method. This can be seen more clearly on a graph:

Another failure of the method is when the graph doesn’t even touch the X-axis at all. So this means that there will be no sign changes resulting in the failure of the graph

Re-arrangement method

Any f(x) function can be rearranged in the form of x = g(x) in multiple ways and any one of those can be used as a basis for iteration for    .This can be used to find a root of the f(x).

As an example I have used to demonstrate it is the f(x) = x 5  + 6x 2  - x + 4

Below is how the graph looks like :

When I re arrange the equation in the form of x= g(x), it turns out to be this:

When I plot this against the y = x graph, this is what the graph looks like:        

This is the visual representation to what happens as x= G(x) gets closer and closer towards the root (staircase).

This is a preview of the whole essay

This shows that after every iteration, I get closer and closer towards the root. After 19 iterations, I found the root of the equation as from then on, it repeats. So the root of the equation x=G(x) against y = x graph.

Failure of Rearrangement method

There are situations where this method doesn’t work. When I re-arrange f(x) into a different x = g(x), it does not converge to the point when I go through the iterations. E.g. I re- arranged f(x) to:

If I plot y = g(x) against y = x, it looks like this:

Looking at this, it implies that the root is between -1 and 0 When I do the iterations starting at the point -1, it turns out like below. The green line is my original equation showing my real root and this clearly shows that the root that is being implied by the y = g(x) graph is wrong. The  iterations are also wrong as they diverge away from the point as well.

If you differentiate the y = g(x) function  and substitute x for the approximation of the root. If that g’(x) < -1 or g’(x) > 1, then we know that the function will not converge to the root but will diverge away.

e.g. Let us take a the last g(x) function possible from my f(x) function.

x= x 5  + 6x 2  + 4 → g(x) = x 5  + 6x 2  + 4

g’(x) = 5x 4  + 12x

From this graph, you can se that the starting x value for the iteration would be -2

G’(x) = 5(-2) 4  + 12(-4)

G’(x) = 32 → 32 > 1 there fore this function should either overflow or diverge away. Here is the proof:

Newton Raphson method

This method works by plotting the f(x) on a graph and visually looking at between what two points the root is (in single units such as 1 or 5 or 9). Then we draw a tangent at that point on the graph. E.g. if the root is between 1 and 2, then you draw a tangent at x= 1 point on the graph. Then we look at where the tangent crosses the X axis and that value will be the new x value on the graph for a tangent. This repeats until we find the root of the equation, (when x value starts repeating or reached enough significant levels). I will attempt to solve f(x) = y = x³−6x²+2x+2

Here’s y = f(x) plotted:

From the graph, we can tell that roots are between (-1 and 0), (0 and 1) and (5 and 6).  So we draw a tangent at x=-1 point on the graph. To do this, we need to find the gradient at that point. To get this, we need to find Y’.

When we substitute in x = -1, we get  . So gradient at that point is 17.

The Equation of the tangent at x = 1 and with grad -7 can be found using the formula below as we know that when x = -1, y = -7 by substituting (-1) in the f(x) : (-1) 3 -6(-x) 2 +2(-1) + 2 = -7

This is the tangent drawn:

From the equation, we can work out where the line crosses the X axis by making

equal to 0:

 -0.58824

So now, our X 1  value is -0.58824 instead of 0. We can generalise this into the formula:

So, now using the formula, we can work out the next X value:

X 1+1  =-0.58824 -  f(X -0.5882 )/f’(X  -0.5882 )

X 2 =  0.58824 –

X 2 = 0.44402

This summarised and finalised in a table from the Software Autograph:

After the fifth iteration, we started repeating the value of X, this means that we have found a root for the equation Y = x 2  - 6x +3

From the table above, you can tell that the root has been achieved completely after 7 iterations as the ∆x is 0. But after the 4 th  iteration, we have reached a point where the x value started repeating due to the fact that this was to 4 decimal places. This is not accurate enough as I want it to be to 5 decimal places. So I used the formula to work out the values in excel. Here are those results:

After the fourth  iteration, we reach the root -0.42362 ± 0.00005

I used use the same method to work out the other two roots. Here are the results for the both these roots. I used excel to work these out.

Root number 2:

Root number 2 is 0.84653 ± 0.00005

Root number 3

Root number 3  is 5.57709 ± 0.00005

Where it does not work

This method does have a flaw. If I choose to work out the roots for the equation f(x) = 4x 2  – 24x + 35 using Newton Raphson method, I find this:

Looking at this, I would expect the roots to be between [3 and 4] and [2 and 3]

I could find the root between 2 and 3 by using the method and it works prefectly to give the value 2.5. But when I try to find the root between 3 and 4, the method fails.

From this, you can see that the tangent has a gradient of 0, therefore does not cross the X axis so we will not get a new X n  Value to carry out the iterations.

Another way of looking at this is using the formula. At the point x = 3, there is a turning point. Therefore the gradient will be zero. This will mean that the gradient function will equal to zero.  If we take a look at the Newton Raphson formula, it’s denominator is the gradient function of f(x). If the gradient function is zero, that will mean that the denominator is 0 and that is an impossible fraction.

Comparisons of the methods:

Now to compare the effectiveness of all the methods, I will attempt to solve exactly same equation, using all three methods. The equation that I will use is y = x 5  + 6x 2  - x + 4

Rearrangement method

I am using the same equation I used for my Re-arrangement above so the results will be exactly the same. So using the Re- arrangement method to solve f(x), I find that the root is - 1.961703555,

- 1.96170 ± 0.00005

This is where we look at between which two points the sign changes from a positive to negative. Similarly to before, I used excel to work out the value of f(x). To start the process, we have to select two points on the graph visual basis. Looking at the graph you can see that the root is between -2 and -1:                                                                          

The root of the function f(x) is -1.961705 ± 0.00005.

Newton Raphson Method

Finally, I am going to use Newton Raphson method to work out the root. I am going to use the inbuilt function in AutoGraph to work it out initially and then I will use Excel to work the root out to more decimal places.

Because this only works out the value to 3 decimal places, I will use the Newton Raphson Formula to work it out to 5 decimal places. I will summarise it in a table like previously

So I have to work out F’ of my equation.

From my graph, I can see that my X 0  will be -2. To work out X 1  I will use the formula above to work out X 1  in steps and will repeat it to find a root to 5d.p and summarise in a table:

Here are the results from the Iterations:

Using this method, the root of the equation is -1.96170. All the methods have given me precisely the same root at 5.d.p.

All three of methods, have situations where they all work, situations where some fail. Out of the three, some are faster at converging to the root that others and some are easier to use than others, especially with the recent technology made available to everyone.

Decimal Search method

This method is by far the easiest and requires least amount of mathematical skill as it requires no complex calculations whatsoever except subbing in X values into the original equation. But with Computers being able to do even that, it makes it even easier to use it. So the only human interaction needed is to look at the results and manually looking at between which two points (limits) there is a sign change and then setting the new limits as such. But regarding the speed of convergence, this is relatively low as it is not completely automated and requires human interaction after each set of results and is very repetitive. Despite this, it is a very effective method as failure chances are relatively very low and the root can be found to many decimal places if the right software is used. With the use of excel, it can be really easy to work out the values of y when you sub in the x values. Writing the formula in a cell and dragging it down so it is easier to calculate the y values for various points of x. Using Autograph is much, much simpler as you have the ability to zoom in on the point where the signs change, i.e. the point on the X-axis, and can do so to many places. This makes it much easier than excel but and is almost accurate as it is because it gives the root to 8+ decimal places and that is accurate.

Fixed point Iteration

This method is probably the hardest method to use out of the three. It involves rearranging a y = f(x) formula into x = g(x), by using any one of the x values in the f(x) formula. This method is slightly complex even on easy formulas, but when we get really complicated when harder formulas are introduced.  This is the hardest bit of the method as the iteration part is really simple, especially with the use of automated software/computer. Most of the mistakes are made in this stage by humans. The downside of the method is that it is the most likely method of all three to fail as there is always a chance that when you rearrange an equation and start the iterative process, it will not converge to the root but it will diverge away. The chance of this happening is quite high. As for the speed of convergence, it would be slower that Newton Raphson because of the extra time needed to re-arrange the equation but it is faster than Decimal search. With the use of Excel, it is very easy to enter the x = g(x) formula in to work out the new value of x and using it for the next iteration by simply dragging down the formula as many cells as you wish (each cell = an iteration). This makes it very easy to work out the repetitive part of the method and makes the speed of convergence much faster. With the use of Autograph, plotting y = g(x) against y = x graph and selecting them both, clicking on  x = g(x) iteration and then selecting a starting point to start the iterations. This makes it even easier than excel but it is less accurate as it only gives the root to 3 decimal places where as excel can be configured to as many decimal places as needed.

Newton Raphson

This method involves working out the gradient function of the graph which is relatively easy even if the equation is complicated. This is the only hard part of the method as the iterative part of the method is very simple with the use of the formula. If automated software/computer is used then it becomes even easier. Regarding the speed the convergence, this is by far the fastest to converge to the root and it is also the method which fails the least out of the three different methods. With the use of excel, it is every easy to write down the formula of the method in one cell and dragging it down as many cells as you wish to (the number of iterations) and it is very easy to use. With the use of AutoGraph Software, It is even easier as you don’t even have to work out the gradient function. All you need to do is just enter y = f(x) formula and then select the graph and select the Newton Raphson iteration and then selecting you X 0  point and click as many times as you want on the arrow (each click = an iteration). This makes it much easier than excel but it is again less accurate as it only gives the root to 3 decimal places where as excel can be configured to as many decimal places as needed.

Overall, all the methods and both the software I used have circumstances where once of them is better than other and circumstances where one completely fails or is much longer. But with ever growing technology, there might already be software that solves the roots of an equation after entering the formula in. But whichever bit of software you use, it is hands down the easier way of finding the roots of an equation.

Teacher Reviews

Here's what a teacher thought of this essay.

An excellent piece of work with no errors giving clear explanations of the use of decimal search, iterations and Newton-Raphson numerical methods to solve equations. 5 stars

Document Details

• Word Count 3460
• Page Count 16
• Level AS and A Level
• Subject Maths

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Teachers’ Unions Are Starting Teacher-Prep Programs. Here’s What to Know

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Teachers’ unions, long defenders of the teaching profession, are now putting their own spin on an often-criticized component: teacher preparation.

The Washington Education Association is in its first year of overseeing a teacher residency program, in which aspiring teachers receive on-the-job training (and a paycheck) while they earn their teaching license. It’s the first—and so far, only—preparation program in which a union is taking the lead role in credentialing teachers, although other state unions are interested in following suit.

Teacher residencies have become increasingly popular at colleges and universities, school districts, and nonprofits, and have been supported by federal grants. Some of those efforts have partnered with teachers’ unions, but the unions had not taken a lead role until now.

But the conditions are ripe for more to follow in WEA’s footsteps: Teacher shortages in certain fields are continuing to plague schools; new sources of federal funding are available ; and teachers’ unions are looking for ways to bolster their own membership.

“Unions have been trying to find ways to engage members, and to innovate, and I think this is a natural avenue to pursue,” said Bradley Marianno, an associate professor of educational policy and leadership at the University of Nevada, Las Vegas. He expects to see more state teachers’ unions adapt the residency model, either by themselves or with a university partner.

After all, this can double as an organizing strategy at a time when teachers’ unions have been hit with legal and legislative challenges to how they recruit and retain members, he said.

“They are in front of these new teachers and can pitch the benefits of the union,” Marianno said.

In the WEA’s residency program, the residents are union members and are covered by a collective bargaining agreement as they earn their teaching certificates.

“Being able to bargain for them, being able to make sure they get all of that bargaining power as a member when they are a resident, I think, is significant,” said Annie Lamberto, the special populations coordinator for WEA who supervises the program.

Also significant: A union-led residency program is designed for and by teachers, she said.

“Everyone that is involved in our program are current members of WEA, and they’re in the classroom,” Lamberto said. “It doesn’t have that disconnect that you can sometimes find in other teacher-prep programs. ... Every single thing we do in our coursework has a direct thread to classroom activities—we don’t make them do anything in our [coursework] that they don’t actually have to perform.”

How Washington’s union-led residency got started

A couple years ago, Washington state schools began relying on record-breaking numbers of emergency substitutes, who aren’t required to have a background in education or a bachelor’s degree.

That pattern was “one of the health indicators that our system is struggling in terms of educator shortages,” said Jim Meadows, the dean of educator career pathways at the WEA.

Chris Reykdal, the state superintendent of public instruction, asked WEA to develop training and support for the emergency substitutes —and create a pathway for teacher certification, Meadows said.

Trade unions in other fields have long run programs to train future workers and supply a pipeline of dues-paying members. And teachers’ unions have long offered professional development, including the clock hours teachers need to maintain their licenses .

“As an organization, we had a strong track record for providing high-quality, relevant, practitioner-led professional learning,” Meadows said.

The WEA received $10.7 million in federal pandemic-relief money from the state, with $6.6 million of it allocated for the union to build and launch a teacher residency program. The first cohort of residents started last year and will graduate in August with their teaching certificate with a special education endorsement.

There are 16 residents in the first cohort who are working in three school districts. The WEA expects to grow to about 30 residents and nine school districts for its second cohort, which will start in June.

The residents are primarily former paraeducators and substitute teachers. They already have bachelor’s degrees, which gives WEA more flexibility: “We do not see our space as being degree-conferring,” Meadows said.

While designing the program, the WEA participated in the National Center for Teacher Residencies’ Residency Design Academy, a consulting service that included a facilitated site visit to another residency to see the work in action.

It was the first time the NCTR worked closely with a union-led teacher residency, said Kathlene Holmes Campbell, the center’s chief executive officer. There’s a lot of opportunity in this space, she said, adding that teachers’ unions are “well positioned” to run such programs, given their experience offering professional learning opportunities to their members.

But in general, it might be beneficial if unions work with an institution of higher education to offer participants a chance to earn a bachelor’s or master’s degree, Campbell said. Creating a pathway to a degree can help attract more diverse candidates.

A more affordable path to the classroom

WEA residents receive a minimum $35,000 salary for the school year, plus benefits. Local teachers’ unions can negotiate for higher salaries, and since the residents are covered by the district’s collective bargaining agreement, they’re eligible for paid time off, including sick days and bereavement leave.

“If you think about the barriers for new teachers, a [collective bargaining agreement] is life-changing in terms of removing those barriers,” Meadows said.

Residents must pay up to $5,000 for the cost of the program, although WEA is looking into ways to reduce that cost, Meadows said. After graduating, residents commit to teaching in the district they trained in for at least three years.

One of the residents, Patrice Madrid, had become a paraeducator in the Federal Way school district in the spring of 2022 after being a stay-at-home mom for eight years. She immediately felt like she had found her calling and wanted to become a full-time teacher—but she didn’t see an immediate path forward.

“I wasn’t sure I would be able to enroll in a traditional program any time soon, because [taking on] significant student loans didn’t seem like the right decision for my family,” she said. “I was pretty disappointed.”

When she learned about WEA’s residency program, it felt like the pieces had fallen into place.

“They have figured everything out that was stopping me, and they have taken that away,” Madrid said. “It was meant to be.”

Rotations in special education settings

In Washington state, as in some others, a special education teaching certificate permits teachers to teach all ages and in all types of special education settings. Yet teachers don’t always get experience in multiple settings during traditional student-teaching.

As in a medical residency, the WEA residency program rotates residents through four nine-week-long rotations in different special education settings, including one in which they work with students with disabilities in general education classes alongside a general education teacher.

“When you become a special ed. teacher, there’s so much variety within that, and you don’t know what you don’t know,” Lamberto said. “We wanted our residents to be able to find not just what they’re good at and what their strengths are, but what they’re passionate about.”

Getting to experience this variety was part of what drew Madrid to WEA’s residency program: “I was nervous about the possibility of training in one type of environment and then becoming a teacher in something completely different,” she said.

Although the residents all have prior classroom experience, the rotations—and corresponding coursework—have ensured that they are getting a wide breadth and depth of knowledge for what it means to be a special education teacher, Lamberto said.

Some of the residents who used to be substitute teachers had never been around students with medical needs before, she said: “They just didn’t really understand what those kids’ needs would be and how they would serve them. ... It was very eye-opening for them.”

Meanwhile, many of the residents who are former paraeducators had asked if they could do their first rotation in the classroom where they used to work. They wanted the comfort of familiarity, Lamberto said—but it backfired.

“For both the mentor and the resident, I think it was too easy to slip back in their role,” she said. “For cohort two, we want to start them off in a place where they’re going to have to stretch.”

How the WEA is making adjustments

In addition to working in classrooms during the day, the residents are taking classes one weeknight per week and on Saturdays, as well as completing assignments.

It’s a demanding schedule, Madrid said. Some residents have had a hard time keeping up with the program’s coursework on top of their full-time job and personal responsibilities.

A challenge for the WEA has been to figure out how to hold residents accountable to the program’s high expectations while still being supportive, Lamberto said.

“That is not something that we have any experience with that a higher ed. organization does,” she said. “What do you do if this student is not doing what they’re supposed to do?”

The WEA has started making tutors—current teachers—available to residents if they need additional support completing their coursework and assignments, Lamberto said.

Another major challenge for WEA will be making sure the program remains financially sustainable after the initial money runs out.

In addition to the pandemic-relief money, the WEA received a $3.2 million grant from the federal State Apprenticeship Expansion Formula program, which the union is using to transition its program into an apprenticeship, which will open up new sources of federal and state funding. The basic structure of the program will remain, but the WEA will make some modifications to align with Washington state’s standards for apprenticeship programs.

Apprentices need a wage progression tied to skill growth, so WEA will have to incorporate at least one pay bump over the program. Secondly, the state requires apprentices to have 2,000 on-the-job training hours. Currently, WEA residents complete around 1,450 eligible hours.

To make up the difference, residents will start the program in January instead of in June, Meadows said, and their rotations will extend to 12 weeks from nine. If the WEA gets approval to transition to an apprenticeship program, this change will take effect in 2025 for the program’s third cohort, which will overlap with the second.

Other unions are paying attention

Other teachers’ unions are watching to see how the WEA’s program continues to evolve.

Monica Byron, the vice president of Education Minnesota, visited Washington state at the start of this year to see the program in action and take notes for how a union-led preparation program could work in the North Star State.

After all, the need is there: Most Minnesota districts have reported having unfilled teacher vacancies, she said.

“We know that we need to do something to find a way to recruit and then retain educators,” Byron said. “Knowing that Washington state has a program that’s seen as successful—we were pretty inspired.”

Education Minnesota formed a committee of educators to discuss what their residency or apprenticeship program might look like. It’s finalizing its recommendations and will present them to the governing board this summer for a vote. If approved, Byron said she hopes to have a program running at the start of the 2025-26 school year.

As union-led preparation programs get underway, though, more systematic research is needed to determine the factors that lead to whether the candidates feel well-prepared to teach, how long they then stay in their jobs, and how their students perform academically, said Jacqueline King, a consultant for research, policy, and advocacy for the American Association of Colleges for Teacher Education.

“We’re still in a period of a lot of experimentation,” she said, referring to alternative programs, both union-led and otherwise. “States are approving programs, and they’re flying blind a little in terms of what the optimal structure is for these kinds of programs.

“I think it’s fantastic that there’s a lot of creative, interesting work being done,” King added. “We need some research to help us understand which of these experiments have really good lessons to teach” the rest of the field.

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Eleven graduating seniors honored with top yale college prizes.

Top row, from left, Maile Somera, Carter Sundown King, Ariana Reichler, Resty Fufunan, Eliza Kravitz, and Xavier Blackwell-Lipkind. Second row, from left, Jimmy Hatch, Jasselene Paz, Jordi Bertrán Ramirez, Matt Brandau, and Andrew Milas. (Photos by Dan Renzetti)

Eleven members of the Yale College Class of 2024 who distinguished themselves in the classroom, on the athletic field, and in their communities were honored with top prizes today in one of Class Day’s most treasured traditions. The prizes were awarded by Yale College Dean Pericles Lewis and other Yale leaders.

The recipients of the top five academic prizes will have the privilege of carrying an official flag or banner during the Yale Commencement procession on May 20. The winner of the Warren Memorial Prize carries the American flag. The winner of the Russell Henry Chittenden Prize carries the Connecticut flag. The winner of the Arthur Twining Hadley Prize carries the Yale College banner. The winners of the Sudler Prize carry the president’s banners. Finally, the winner of the Alpheus Henry Snow Prize carries the Yale University banner.

The names of the prizewinners, and their citations, are listed below, in the order of their presentation.

The Nellie Pratt Elliot Award

Awarded to a senior woman whose excellence on the field of play, and in her life at Yale, best represents the highest ideals of American sportsmanship and Yale tradition .

MAILE SOMERA, Pierson College

“ Maile Somera is the team captain of the 3-time Ivy League championship volleyball team, which won the Ivy League tournament this year for the second year in a row. She was the 2023 Ivy League Defensive Player of the Year and is 2nd All-time at Yale in number of aces served to opponents — who watched the balls whiz by, unable to return them. And, Maile is graduating with distinction in her Architecture major!

“ In recognition of her record as a fearless opponent, a consummate leader, a fine student, and a selfless example, Yale College is proud to award the Nellie Elliott Award to Maile Somera.”

The William Neely Mallory Award

Awarded to a senior man whose excellence on the field of play, and in his life at Yale, best represents the highest ideals of American sportsmanship and Yale tradition .

MATT BRANDAU, Benjamin Franklin College

“ Matt Brandau is a 3-time NCAA tournament participant, a 3-time All-American, the 2023 New England Player of the Year, the All-time leading scorer in Yale men’s lacrosse and the All-time leading scorer in Ivy League history, and the 2024 Ivy League Player of the Year.

“ For his truly extraordinary athletic accomplishments, and for the qualities that make him a quintessential teammate, and the epitome of the scholar-athlete ideal, Yale College takes pride in honoring him with the William Neely Mallory Award.”

The Nakanishi Prize

Awarded to two graduating seniors who, while maintaining high academic achievement, have provided exemplary leadership in enhancing race or ethnic relations at Yale College .

RESTY FUFUNAN, Trumbull College

“ Resty Fufunan has embodied the Nakanishi Prize through his activism, grounded in equal parts of listening, critical thinking, and hard, pragmatic work. He has been a community builder through his many roles on campus, among them First-Year Counselor, Co-Head Counselor for Camp Yale’s Cultural Connections program, and choreographer and dancer for DanceWorks. Thanks to him, this community has become more open and welcoming.

“ Resty has also been a pillar of the Asian American community. He has served as a first-year liaison; President of the Filipino student organization, Kasama; student co-head of the Asian American Cultural Center; and co-moderator of the Asian American Students Alliance. He has coordinated an inter-group trip to Washington, D.C., to protest the Supreme Court's oral arguments in cases about affirmative action, in the process building a foundation for future coalitions on ethnic relations and advocacy.

“ Resty’s two majors — Ethnicity, Race, and Migration and Data Science — have enabled him to promote ethnic and racial relations by focusing on the policy intersections between data and social justice. He has also applied that knowledge through internships, analyzing census data to prepare a national survey of Asian American voters. Now that he is graduating, he will continue his studies in China next year as the recipient of a Richard U. Light Fellowship.

“ For his transformational work advancing ethnic and race relations, and his unforgettable impact on ethnic relations in our community, Yale College is proud to bestow the Nakanishi Prize upon Resty Fufunan.”

JASSELENE PAZ, Silliman College

“ Jasselene Paz is a community builder who has forged partnerships, re-examined histories, and created new communities for Yale’s campus culture. She has dedicated herself to making Yale a place where multiple and intersecting identities — around race, gender, sexuality and ethnicities — can take root, bloom, and flourish.

“ Jasselene is the founder and president of Central Americans for Empowerment; a radio host with WYBCx Yale Radio; a co-head counselor for Camp Yale’s Cultural Connections program; a Peer Liaison for La Casa Cultural; a Silliman Latine Affinity Group Co-Founder & Event Coordinator; a dancer and leader in the dance groups Sabrosura and Rhythmic Blue; and a Community Consent Educator.

“ An Ethnicity, Race, & Migration major and a Human Rights scholar, Jasselene has brought the theory and practice of community building to a global context. She has studied at Yonsei University, South Korea, as a recipient of a Richard U. Light Fellowship, and also in Cartagena, Colombia, working with community members and organizations to understand how the country’s armed conflict and its subsequent 2016 Peace Accords have affected Black, Indigenous, Brown, urban, rural, and gender- marginalized communities.

“ For her many contributions to Yale’s campus culture, Yale College is honored to bestow the Nakanishi Prize upon Jasselene Paz.”

The James Andrew Haas Prize

Awarded to that member of the senior class in Yale College whose breadth of intellectual achievement, strength of character, and fundamental humanity shall be adjudged by the faculty to have provided leadership for his or her fellow students, inspiring in them a love of learning and concern for others .

JIMMY HATCH, Timothy Dwight College

“ Jimmy Hatch entered Yale at the age of 52, after a long and distinguished career as a Navy SEAL in the United States military. He graduates with a degree in Humanities, concluding an undergraduate career that began in Directed Studies — he was the first Eli Whitney student to enroll in the program — and culminating in a senior thesis exploring the interplay between literature and lived experience, with a particular focus on depictions of combat in classical Greek, Roman, and Italian texts.

“ Jimmy’s extracurricular life on campus has been rich and varied, ranging from his partnership with the Jackson School in creating an “After Action’ class immediately following the U.S. exit from Afghanistan to his collaboration with the Yale University Art Gallery on a ‘Public Plato’ project, for which he recently hosted a conversation with Dean Tamar Gendler on ‘Ancient Thinking and the Modern Human.’

“ Jimmy is invested in the life of the mind, in the creation of community, in often winding pursuit of light and truth. For the depth and breadth of his commitments, Yale College is proud to bestow the James Andrew Haas Memorial Prize upon James Hatch.”

The Warren Memorial Prize

Awarded to the senior majoring in the humanities who ranks highest in scholarship .

ELIZA KRAVITZ, Morse College

“ Eliza Kravitz graduates summa cum laude with distinction in her History major. Elected to Phi Beta Kappa in her junior year, she is also a member of the National Collegiate Hispanic Honor Society. Eliza is a recipient of the Yale Review of International Studies Acheson Prize for her essay in international affairs. This year, she also received the Carlos R. Morena Prize, given annually to the best student paper focusing on the field of Latinx Studies.

“ Outside the classroom, Eliza has made a significant impact in the surrounding communities of Connecticut. As a first-year student, she helped students incarcerated in the Manson Youth Institution pass the GED. She has also volunteered as a tax preparer with organizations that assist under-served families in preparing their taxes, and she has worked as a Spanish interpreter for the New Haven Legal Assistance Association.

“ In recognition of her extraordinary scholarly achievements, Yale College is proud to award the Warren Memorial High Scholarship Prize this year to Eliza Rose Kravitz.”

The Arthur Twining Hadley Prize

Awarded to the senior in Yale College majoring in the social sciences who ranks highest in scholarship .

ARIANA REICHLER, Trumbull College

“ Ariana Reichler was elected to Phi Beta Kappa in her junior year and graduates summa cum laude with distinction in her major, Cognitive Science, and a certificate in Education Studies. Her academic pursuits have been broad and interdisciplinary, with a steady focus on cognitive development.

“ As a member of theClinical Affective Neuroscience & Development Lab since her sophomore year, Ariana’s research has focused on child and adolescent development and its intersection with mental health, specifically examining mechanisms linking childhood adversity exposure with risk for posttraumatic stress disorder.

“ Outside of the classroom, Ariana’s extracurricular commitments have often focused on children, too. From serving as a Community Mental Health Fellow with Dwight Hall and the Connecticut Mental Health Center Foundation to volunteering as a swim instructor to local children with special needs or disabilities, Ariana has made a significant impact in the broader New Haven community.

“ For her exceptional scholarship, Yale College proudly awards the Arthur Twining Hadley Prize to Ariana Reichler.”

The Russell Henry Chittenden Prize

Awarded to the senior majoring in the natural sciences or in mathematics who ranks highest in scholarship .

ANDREW MILAS, Grace Hopper College

“ Andrew Milas graduates summa cum laude, with distinction in both his majors, Computer Science and Mathematics. He has distinguished himself as one of the most outstanding scholars in his fields of study, showing extraordinary talent in his demanding coursework, much of it at the graduate level, and far exceeding the requirements for the major or the undergraduate degree. Whether studying economics or artificial intelligence, deep learning theory or statistics and data science, he has done, in the words of one of his instructors, ‘spectacularly.’

“ Andrew is the recipient of numerous awards and prizes, among them a first-place win in the Jane Street Electronic Trading Challenge Final Hour. At the William Lowell Putnam Mathematical Competition, he and his teammates took fifth place in the 2022 Putnam Exam, he received an honorable mention, and Yale’s Department of Mathematics praised Andrew and his team as placing Yale among ‘the top five for the first time since 1991,’ a first in over three decades.

“ For his exceptional scholarship and future promise in his field, Yale College proudly awards the Russell Henry Chittenden Prize to Andrew Milas.”

The Louis Sudler Prize

Awarded to two seniors for excellence in the performing or creative arts .

CARTER SUNDOWN KING, Pauli Murray College

“ Carter Sundown King graduates with distinction in his major, Theater, Dance, and Performance Studies. Admired by faculty and students alike, he is not only a costume designer who raises the standard of every production he joins; he also has a deep respect for the materials, histories, and bodies with which he works. Moving between rigorous research and ‘intuitive leaps of the imagination’ his costumes demand to be seen and understood.

“ An enrolled member of the Oneida Nation, Carter King's design practice centers his Oneida experience, drawing on lessons in creativity, community, integrity, and ingenuity he learned while being raised on the Oneida Nation Reservation in Wisconsin. If you find yourself anywhere in his vicinity, you cannot help but notice him. He wears his Nation and his art form everywhere he goes, teaching us all how to carry elements of what we love with us, on our very bodies.

“ For his unwavering dedication to the highest standards of his art form, Yale College is honored to award the Louis Sudler Prize for Excellence in the Performing and Creative Arts to Carter Sundown King.”

JORDI BERTRÁN RAMIREZ, Trumbull College

“ Jordi Bertrán Ramirez of Trumbull College graduates with a double major in Theater, Dance, and Performance Studies and in History of Science, Medicine, and Public Health. In his doubled studies he has investigated popular and political cultures of information, identity, and representation, using not only classrooms and computers but also stages, voices, and bodies to explore urgent questions.

“ Anyone who has seen Jordi on stage (in his over 30 productions at Yale), knows that he has ‘It’ — the ‘it’ of a theater actor’s magnetic force that compels audiences to watch and listen; the ‘it’ of an actor’s grace, charisma, and artistic intelligence; the ‘it’ that Professor Joseph Roach has called ‘easy to perceive but hard to define, possessed by abnormally interesting people.’ Jordi is not only such a person; he brings his own interests and ethics alive in his performances; he plays the stage as a virtuoso plays the piano; he channels his talent into projects that matter — to him, to his community, and to our changing and challenging world.

“ For his luminous accomplishments in theatrical performance, Yale College is honored to award the Louis Sudler Prize for Excellence in the Performing and Creative Arts to Jordi Bertrán Ramirez.”

The Alpheus Henry Snow Prize

Awarded to the senior who through the combination of intellectual achievement, character, and personality, shall be adjudged by the faculty to have done most for Yale by inspiring in his or her classmates an admiration for the traditions of high scholarship .

XAVIER BLACKWELL-LIPKIND, Davenport College

“ Xavier Blackwell-Lipkind graduates summa cum laude and with distinction in his major, Comparative Literature. Elected to Phi Beta Kappa in his junior year and serving as its president in his senior year, Xavier has also completed an Advanced Language Certificate in French. Next year, he will pursue a Masters in Comparative Literature at Oxford as a Marshall Scholar.

“ Xavier has devoted much of his undergraduate years exploring languages. He has received numerous awards for his work in Spanish, French, and Portuguese, including the Scott Prize for the best essay in French and the Bildner Prize for ‘an outstanding essay in Spanish on any subject in Latin American Literature and/or Culture.’ The English department has also recognized Xavier, twice awarding him the John Hubbard Curtis Prize ‘for love of the English language and facility in writing’ as well as the sophomore C. Wyllys Betts Prize and the Elmore A. Willets Prize for Fiction. His love of languages continues with ongoing study of Italian and Amharic.

“ Awards from outside Yale have also been bestowed on Xavier for his fiction and non-fiction prose stories, which have been published in literary journals such as The Threepenny Review, The Drift, Gulf Coast, West Branch, and Brevity, to name just a few. He won the Editors’ Prize in Prose from the Copper Nickel.

“ Xavier’s deep belief in the value of language and literature ties directly to his enthusiastic service as a mentor to others. On campus, Xavier tutors students in French and Portuguese. In the New Haven community, he uses his talents and expertise to serve as a volunteer translator and interpreter for immigration nonprofits and asylum attorneys.

“ Xavier has contributed to the life of Yale through his writing as well. Since May 2023, he has served as Editor-in-Chief of the Yale Literary Magazine, having previously occupied the roles of managing editor and literary editor. In addition, he has worked as a staff writer for the Yale Daily News Magazine and as a reporter for the YDN. Somehow, he has also played the viola in the Yale Symphony Orchestra for two years.

“ For his remarkable achievements and his promise for the many more to come, Yale College takes great pleasure in bestowing the Alpheus Henry Snow Prize upon Xavier Blackwell-Lipkind.”

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